Chapters 8-9 introduced the Solow model: capital accumulation drives output toward a steady state, but long-run growth in output per worker requires exogenous technological progress. This chapter asks: where does technological progress come from? If ideas drive growth, and ideas are produced by people making purposeful decisions, then growth itself is endogenous.
We begin by formalizing the Solow model's insights through the Ramsey-Cass-Koopmans framework (optimal saving), then build toward endogenous growth: the AK model, Romer's variety-expansion model, and Aghion-Howitt's Schumpeterian creative destruction.
Prerequisites: Chapters 8-9 (Solow model). Mathematical prerequisites: dynamic optimization, phase diagrams, differential equations.
Named literature: Ramsey (1928); Cass (1965); Koopmans (1965); Diamond (1965); Romer (1986, 1990); Lucas (1988); Aghion & Howitt (1992); Mankiw, Romer & Weil (1992).
The Solow model assumes a fixed saving rate $s$. The Ramsey model endogenizes saving by having a representative household choose consumption and saving to maximize lifetime utility.
Preferences: An infinitely-lived representative household with CRRA utility:
The parameter $\sigma$ is the coefficient of relative risk aversion (inverse of the intertemporal elasticity of substitution, IES $= 1/\sigma$). Technology: $y = f(k)$ in per-effective-worker terms with CRS. Capital depreciates at rate $\delta$; population grows at $n$; TFP grows at $g$.
FOCs: $\lambda = c^{-\sigma}$ (Eq. 13.3) and $\dot{\lambda}/\lambda = \rho - [f'(k) - (n + g + \delta)]$ (Eq. 13.4).
This is the Keynes-Ramsey rule. Consumption grows when the marginal product of capital exceeds the effective discount rate.
At steady state: $f'(k^*) = \rho + \delta + \sigma g$ (modified golden rule) and $c^* = f(k^*) - (n + g + \delta)k^*$.
The Ramsey economy always under-accumulates relative to the golden rule ($k^* < k_g$) because impatient households consume too much today. Dynamic inefficiency is impossible.
Figure 13.1. Ramsey phase diagram. The vertical blue line is the $\dot{c}=0$ locus; the hump-shaped red curve is the $\dot{k}=0$ locus. The green dashed line is the saddle path. Arrows show dynamics in each region. Adjust parameters and click to launch trajectories.
With $f(k) = k^{1/3}$, $\rho = 0.04$, $\delta = 0.05$, $g = 0.02$, $\sigma = 2$:
$\dot{c} = 0$: $f'(k^*) = (1/3)k^{*-2/3} = 0.04 + 0.05 + 2(0.02) = 0.13$
$k^* = [(1/3)/0.13]^{3/2} = 4.11$, $c^* = (4.11)^{1/3} - (0.09)(4.11) = 1.23$
Starting from $k_0 = 1 < k^* = 4.11$ (parameters from Example 13.1), characterize the saddle-path dynamics.
Step 1: At $k_0 = 1$, $f'(1) = 1/3 > 0.13 = \rho + \delta + \sigma g$, so $\dot{c}/c > 0$: consumption is rising.
Step 2: On the saddle path, $c_0$ must jump to the value where the trajectory converges to $(k^*, c^*)$. If $c_0$ is too high, consumption grows too fast, capital is depleted, and the economy hits $k = 0$. If $c_0$ is too low, capital accumulates forever, violating transversality.
Step 3: Along the saddle path, both $k$ and $c$ rise monotonically toward the steady state. The economy grows rapidly at first (high $f'(k)$) and decelerates as $k \to k^*$.
Key insight: The saddle path is the unique rational-expectations equilibrium. Forward-looking households must select $c_0$ perfectly to land on it.
The Ramsey model assumes an infinitely-lived representative household. Diamond (1965) replaces this with overlapping generations: each agent lives two periods (young and old), and at every date a new generation is born while the oldest generation dies. This seemingly small change has profound consequences — most importantly, the economy can over-accumulate capital.
Budget constraints: A young agent born at $t$ earns wage $w_t$, consumes $c^y_t$, and saves $s_t$. When old at $t+1$, the agent consumes the return on savings:
The agent maximizes lifetime utility:
where $\beta = 1/(1+\rho)$ is the discount factor (consistent with the Ramsey notation: $\rho$ is the rate of time preference). Substituting the budget constraints and taking the first-order condition with respect to $s_t$:
This is the OLG Euler equation. It has the same economic logic as the Ramsey Euler equation (Eq. 13.5): the marginal utility cost of saving one more unit when young equals the discounted marginal utility benefit of consuming the return when old. With CRRA utility $u(c) = c^{1-\sigma}/(1-\sigma)$, the optimal saving function $s_t = s(w_t, r_{t+1})$ has an ambiguous response to the interest rate — the substitution effect (save more) and income effect (can save less and still afford the same old-age consumption) work in opposite directions.
Capital accumulation. All saving by the young becomes next period's capital. With population growing at rate $n$ ($L_{t+1} = (1+n)L_t$), the per-worker capital stock evolves as:
With Cobb-Douglas production $f(k) = k^\alpha$ and log utility ($\sigma = 1$), optimal saving is $s_t = \beta w_t / (1+\beta)$ (the interest rate drops out). Substituting $w_t = (1-\alpha)k_t^\alpha$:
This is a concave difference equation in $k$ with a unique positive steady state:
Dynamic efficiency vs. inefficiency. The golden rule capital $k_g$ satisfies $f'(k_g) = n$ (in this simplified model without depreciation or TFP growth). If $k^* > k_g$, the economy has too much capital: the return on saving is below the population growth rate ($r < n$), and every generation could be made better off by saving less.
Why Ramsey avoids this: In the Ramsey model, the transversality condition (Eq. 13.6) rules out paths where capital grows without bound. The infinitely-lived household would never save past the point where $r < \rho + n + g$ because doing so violates optimality. In the OLG model, no single agent cares about the infinite future. The young save based on their own two-period calculus. If agents are sufficiently patient ($\beta$ high) and population growth is slow ($n$ low), aggregate saving can overshoot the golden rule.
The AMSZ test (Abel, Mankiw, Summers & Zeckhauser, 1989): If aggregate capital income exceeds aggregate investment, the economy is dynamically efficient. For all major OECD economies, capital income > investment, confirming dynamic efficiency. This is reassuring — real economies do not appear to over-accumulate.
Kaelani, with a young population ($n \approx 0.02$) and low saving rates (recall $s = 0.15$ from the Solow comparison in Chapter 8), is firmly in the dynamically efficient region — its problem is under-accumulation, not over-accumulation. Weak property rights (Chapter 12) and limited financial intermediation discourage the saving that would raise $k$ toward the golden rule.
Figure 13.2. OLG capital dynamics. The blue curve is the law of motion $k_{t+1} = g(k_t)$; the gray line is the 45° line. Their intersection is the steady state $k^*$. The dashed vertical line marks the golden rule $k_g$. When $k^* > k_g$ (red shading), the economy is dynamically inefficient. Drag the sliders: patient agents or slow population growth push $k^*$ rightward, potentially past $k_g$.
Example 13.3 — Diamond Model: Steady State and Dynamic Efficiency
Setup: Log utility ($\sigma = 1$), Cobb-Douglas production $f(k) = k^{1/3}$ ($\alpha = 1/3$), $\beta = 0.5$, $n = 0.02$, $\delta = 0$.
Step 1 (Euler equation): With log utility, $1/c^y_t = \beta(1+r_{t+1})/c^o_{t+1}$. Substituting the budget constraints: $s_t = \beta w_t / (1+\beta) = 0.5 w_t / 1.5 = w_t/3$.
Step 2 (Law of motion): $k_{t+1} = \frac{0.5 \times (2/3)}{1.5 \times 1.02} k_t^{1/3} = \frac{1/3}{1.53} k_t^{1/3} \approx 0.2178\, k_t^{1/3}$.
Step 3 (Steady state): $k^* = (0.2178)^{3/2} \approx 0.1017$. Output: $y^* = (0.1017)^{1/3} \approx 0.467$.
Step 4 (Golden rule): $f'(k_g) = n \Rightarrow (1/3)k_g^{-2/3} = 0.02 \Rightarrow k_g = (1/(3 \times 0.02))^{3/2} = (16.67)^{1.5} \approx 68.04$.
Step 5 (Comparison): $k^* = 0.102 \ll k_g = 68.04$. The economy is far below the golden rule: dynamically efficient. The marginal product $r = f'(k^*) = (1/3)(0.102)^{-2/3} \approx 1.53 \gg n = 0.02$.
Key insight: With $\beta = 0.5$ (moderate patience), agents do not save enough to overshoot the golden rule. Dynamic inefficiency requires much higher $\beta$ (try $\beta > 0.95$ in the interactive figure above).
The Solow and Ramsey models predict that growth in output per worker eventually ceases (absent exogenous $g$) because of diminishing returns to capital. The AK model eliminates diminishing returns.
where $A$ is a constant and $K$ is interpreted broadly (physical + human capital + knowledge).
Growth is perpetual and proportional to the saving rate. There is no steady state — no convergence. Policy (higher $s$) permanently affects the growth rate, not just the level.
Figure 13.3. Solow vs. AK model. In Solow (left), a higher saving rate shifts the steady state up — a level effect. In the AK model (right), a higher saving rate raises the growth rate permanently. Drag the slider to compare.
Paul Romer's key insight: ideas are non-rival. A design for a microchip, once created, can be used by any number of firms simultaneously. Non-rivalry implies increasing returns to scale. Romer resolved the incompatibility with competition by introducing monopolistic competition — innovators earn temporary monopoly profits through patents.
New varieties are created by researchers ($L_A$), building on existing knowledge ($A$). On the balanced growth path:
Scale effects: A larger economy (more potential researchers) grows faster. This is both the model's prediction and its most debated feature.
Figure 13.4. Romer's ideas production. The left axis shows the growth rate of ideas as a function of R&D labor share. The right panel shows the scale effect: larger economies (more total labor) produce more growth for the same R&D share. Drag the slider to explore.
An economy has $L = 1{,}000{,}000$ workers, R&D labor share $L_A/L = 0.05$, and R&D productivity $\delta_A = 0.0004$.
Step 1: Number of researchers: $L_A = 0.05 \times 1{,}000{,}000 = 50{,}000$.
Step 2: Growth rate of ideas: $g_A = \delta_A L_A = 0.0004 \times 50{,}000 = 20$ ... but we need to interpret units. With $\delta_A = 0.0004$ per researcher, $g_A = 0.0004 \times 50{,}000 = 20$? That gives 2000%/year. Let us re-calibrate: $\delta_A = 0.00004$, then $g_A = 0.00004 \times 50{,}000 = 2.0$, i.e., 2.0%/year.
Step 3: On the balanced growth path, $g_Y = g_A = 2.0\%$/year. Doubling time: $\ln 2 / 0.02 = 34.7$ years.
Step 4 (scale effects): If population doubles to 2M with the same R&D share, $L_A = 100{,}000$, and $g_A = 4.0\%$/year. The Romer model predicts that larger economies grow faster — a prediction that has been challenged empirically.
In the Romer model, derive the balanced growth path (BGP) where all growth rates are constant.
Step 1: Ideas production: $\dot{A}/A = \delta_A L_A$. On the BGP, $L_A$ is constant (fixed fraction of labor), so $g_A = \delta_A L_A$ is constant.
Step 2: Final goods production: $Y = A^\phi K^\alpha L_Y^{1-\alpha}$ (where $\phi$ captures the ideas externality). On the BGP, $g_Y = \phi g_A + \alpha g_K + (1-\alpha)g_{L_Y}$.
Step 3: Capital accumulates from saving: $g_K = sY/K - \delta$. On the BGP, $g_K = g_Y$ (constant $K/Y$ ratio).
Step 4: Substituting $g_K = g_Y$ and $g_{L_Y} = n$: $g_Y = \phi g_A + \alpha g_Y + (1-\alpha)n$, so $g_Y(1-\alpha) = \phi g_A + (1-\alpha)n$, giving $g_Y = \frac{\phi}{1-\alpha}g_A + n$.
Step 5: Per capita growth: $g_{Y/L} = g_Y - n = \frac{\phi}{1-\alpha}\delta_A L_A$. Growth in living standards is proportional to R&D effort.
Aghion and Howitt (1992) model growth through creative destruction. Innovation follows a Poisson process; each innovation improves quality by factor $\gamma > 1$.
Two opposing externalities: the business-stealing effect (innovator captures incumbent's rents — excessive incentive) and the knowledge spillover effect (innovator doesn't capture benefit to future innovators — insufficient incentive). Empirical evidence suggests spillovers typically dominate, justifying R&D subsidies.
Each bar represents an industry's current quality level on the ladder. Click Step to advance one innovation round: industries that receive an innovation see their quality jump by factor $\gamma$, while the displaced incumbent flashes red. Higher R&D intensity means more industries innovate per step.
Figure 13.5. Aghion-Howitt quality ladder. Each bar is an industry; height is log quality level. Click Step to trigger an innovation round — innovating industries jump up (blue) while displaced incumbents flash red. Higher R&D intensity increases the share of industries that innovate each period, raising the aggregate growth rate. Observe how creative destruction drives growth.
In the Aghion-Howitt model with arrival rate $\lambda \phi(n) = \lambda n$ (linear in R&D labor $n$), quality step $\gamma = 1.2$, and interest rate $r = 0.05$:
Step 1: Growth rate: $g = \lambda n \ln\gamma$. With $\lambda = 0.5$ and $n = 0.10$: $g = 0.5 \times 0.10 \times \ln(1.2) = 0.5 \times 0.10 \times 0.182 = 0.0091$ or 0.91%/year.
Step 2: The social planner maximizes welfare considering that each innovation creates a knowledge spillover for future innovators. The private innovator ignores this externality.
Step 3: Business-stealing effect: the innovator captures the incumbent's rents (excess private incentive = $\pi_{old}$). Knowledge spillover: the innovator raises the quality frontier for future innovators (insufficient private incentive).
Step 4: If the spillover dominates (typical case), the social optimum has $n^* > n_{market}$, justifying R&D subsidies. If business-stealing dominates, the market over-invests in R&D.
Unconditional convergence fails: many of the world's poorest countries in 1960 remain poorest today. Conditional convergence succeeds: controlling for steady-state determinants, poorer countries grow faster. Speed of convergence: ~2%/year (half-life ~35 years).
Figure 13.6. Convergence visualizer. Two countries start at different capital stocks (k0=1 in blue, k0=8 in red) but share the same fundamentals. Both converge to the same steady state. Adjusting institutional quality A shifts the shared steady state. Watch the animated convergence paths.
MRW added human capital ($h$) to the Solow model:
MRW showed the augmented Solow model explains ~80% of cross-country income variation — a dramatic improvement over the basic model (~60%).
Figure 13.7. MRW-style regression: log GDP per capita vs. log investment rate, colored by human capital (schooling). Countries with higher human capital (larger, greener dots) tend to be richer. The fitted line shows the strong positive relationship between investment and income. Hover for country details.
TFP growth (the Solow residual) accounts for a large share of growth in advanced economies. Capital accumulation alone cannot drive sustained growth.
Between 1966 and 1990, South Korea's GDP grew at 10.3%/year. Decompose this using growth accounting.
Data: Capital growth $g_K = 13.7\%$/year. Labor growth $g_L = 6.4\%$/year (including quality adjustment). Capital share $\alpha = 0.35$.
Step 1: Capital contribution: $\alpha \cdot g_K = 0.35 \times 13.7\% = 4.8\%$.
Step 2: Labor contribution: $(1-\alpha) \cdot g_L = 0.65 \times 6.4\% = 4.2\%$.
Step 3: TFP residual: $g_A = g_Y - \alpha g_K - (1-\alpha)g_L = 10.3\% - 4.8\% - 4.2\% = 1.3\%$.
Interpretation: Factor accumulation (capital + labor) accounts for 87% of Korean growth. TFP accounts for only 13%. This led to the "perspiration vs. inspiration" debate: was the Asian miracle driven by brute-force accumulation (Young, 1995) or genuine productivity gains?
Mankiw, Romer, and Weil (1992) estimate the augmented Solow model:
$$\ln(Y/L) = \text{const} + \frac{\alpha}{1-\alpha-\beta}\ln s_K + \frac{\beta}{1-\alpha-\beta}\ln s_H - \frac{\alpha+\beta}{1-\alpha-\beta}\ln(n+g+\delta)$$
Step 1: With $\alpha = 1/3$ and $\beta = 1/3$: the coefficient on $\ln s_K$ is $\frac{1/3}{1/3} = 1.0$; on $\ln s_H$ is $\frac{1/3}{1/3} = 1.0$; on $\ln(n+g+\delta)$ is $-\frac{2/3}{1/3} = -2.0$.
Step 2: A country that doubles its physical investment rate ($s_K$) increases steady-state income by $\exp(1.0 \times \ln 2) = 2.0$, i.e., 100%.
Step 3: A country that doubles its human capital investment ($s_H$) also doubles income. Human capital is as important as physical capital.
Step 4: The augmented model (R$^2 \approx 0.78$) dramatically outperforms the basic Solow model (R$^2 \approx 0.59$). Adding human capital resolves the "too high" predicted convergence speed of the basic model.
Solow's 1987 quip: "You can see the computer age everywhere but in the productivity statistics."
Despite massive investment in information technology during the 1970s and 1980s, measured TFP growth in the United States actually slowed — from 1.5%/year in 1948–73 to 0.3%/year in 1973–95. Computers were transforming offices, factories, and daily life, yet the growth statistics showed nothing.
Three explanations emerged: (1) Measurement error — national accounts struggled to capture quality improvements in new goods and services. How do you measure the productivity gain from email replacing postal mail? (2) Implementation lags — general-purpose technologies require complementary investments (reorganization, training, new business processes) that take decades. Electricity showed a similar pattern: invented in the 1880s, productivity gains visible only in the 1920s. (3) Redistribution, not creation — some IT investment simply shifted rents between firms without raising aggregate productivity.
Resolution: Productivity surged in the late 1990s (TFP growth jumped to 1.4%/year in 1995–2004), concentrated in IT-using sectors like retail and wholesale trade. The productivity paradox was real but temporary — the computer age eventually showed up in the statistics, vindicating Solow's framework while highlighting the limits of growth accounting in real time.
Goldman Sachs published a research note estimating that generative AI could automate 300 million full-time jobs globally. The Economist ran it as a cover story. Elon Musk posted that "AI will make most jobs pointless." But every previous automation panic — from the Luddites to the ATM scare — was wrong. Romer's model says more ideas mean more output and more demand for labor. Is this time actually different?
IntermediateYou now have endogenous growth — Romer, Aghion-Howitt, creative destruction, and the empirics of convergence. Ideas explain sustained growth. But that raises the hardest question of all.
Romer (1990): ideas are non-rival. More researchers producing more ideas means faster growth. The steady-state growth rate depends on R&D investment. The AK model shows that if returns to broad capital (including human capital) don't diminish, growth can persist without exogenous technological change. Growth accounting confirms that TFP — the catchall for ideas and efficiency — accounts for 50–70% of cross-country income differences. Countries that invest in R&D and have institutions that incentivize innovation grow faster.
Scale effects are the Achilles heel: Romer predicts that larger populations produce faster growth, which doesn't hold (Jones, 1995). Semi-endogenous growth fixes this but makes the growth rate depend on population growth, not policy — a depressing result. More fundamentally: if ideas are non-rival, why don't poor countries just copy existing ideas? The technology frontier should diffuse freely. The fact that it doesn't suggests the barrier is not ideas per se but something about the social and institutional environment — extractive institutions, weak property rights, corruption, low human capital. Endogenous growth theory tells you what the engine is but not why some countries have it and others don't.
Aghion-Howitt's creative destruction and directed technical change models enriched the framework. The frontier moved from asking about technology creation to asking about the barriers to technology adoption. Why can't Nigerian farmers use the same seeds as American farmers? Why don't Indian manufacturers adopt the same management practices as Japanese ones? The answer, increasingly, is not that the technology is unavailable but that institutions, infrastructure, and human capital prevent adoption.
Ideas are the proximate engine of growth — this is Romer's lasting contribution. But "ideas" is itself endogenous to institutions, incentives, and social structures. Growth theory tells you what drives growth (ideas, R&D, creative destruction) but not why some countries build innovation systems and others don't. The model works beautifully for the OECD; it's incomplete for the global South. That incompleteness is not a failure of the theory — it's a signpost pointing toward institutions.
If technology is globally available but unevenly adopted, the barrier must be institutional. Chapter 18 (§18.3–18.4) introduces the Acemoglu-Johnson-Robinson framework: extractive institutions prevent technology adoption, inclusive institutions enable it. The settler mortality instrument provides causal evidence. But "fix the institutions" is easier said than done — and the China miracle challenges the entire extractive/inclusive binary.
Goldman Sachs published a research note estimating that generative AI could automate 300 million full-time jobs globally. The Economist ran it as a cover story. Elon Musk posted that "AI will make most jobs pointless." But every previous automation panic — from the Luddites to the ATM scare — was wrong. Romer's model says more ideas mean more output and more demand for labor. Is this time actually different?
IntermediateKaelani (GDP = \$10B, pop = 5M, s = 0.15) spends 0.5% of GDP on R&D: ~500 researchers. In the Romer framework, this may be insufficient for meaningful frontier innovation.
But three factors help: (1) Knowledge diffusion — ideas are non-rival, so Kaelani can adopt technologies from abroad. (2) Specialization — focus R&D on niches like tropical agriculture. (3) Institutions — the Chapter 18 reforms raise TFP by reducing corruption.
Growth accounting (2010-2025): GDP growth 4.0%/year = capital accumulation (2.0%) + labor growth (1.0%) + TFP growth (1.0%). The 1% TFP growth is driven by institutional reform and technology adoption, not frontier innovation.
| Label | Equation | Description |
|---|---|---|
| Eq. 13.1 | $\max \int e^{-\rho t}u(c)dt$ s.t. $\dot{k} = f(k)-c-(n+g+\delta)k$ | Ramsey household problem |
| Eq. 13.5 | $\dot{c}/c = (1/\sigma)[f'(k) - \rho - \delta - \sigma g]$ | Euler equation |
| Eq. 13.6 | $\lim_{t\to\infty} \lambda(t)k(t)e^{-\rho t} = 0$ | Transversality condition |
| Eq. 13.9 | $u'(c^y_t) = \beta(1+r_{t+1})u'(c^o_{t+1})$ | OLG Euler equation |
| Eq. 13.11 | $k_{t+1} = \frac{\beta(1-\alpha)}{(1+\beta)(1+n)}k_t^\alpha$ | OLG law of motion (log utility) |
| Eq. 13.12 | $k^* = \left[\frac{\beta(1-\alpha)}{(1+\beta)(1+n)}\right]^{1/(1-\alpha)}$ | OLG steady state |
| Eq. 13.13 | $Y = AK$ | AK production function |
| Eq. 13.14 | $g_Y = sA - \delta$ | AK growth rate |
| Eq. 13.15 | $\dot{A} = \delta_A L_A A$ | Romer ideas production |
| Eq. 13.16 | $g_A = \delta_A L_A$ | Romer balanced growth rate |
| Eq. 13.18 | $g = \lambda\phi(n)\ln\gamma$ | Aghion-Howitt growth rate |
| Eq. 13.19 | MRW augmented Solow regression | Cross-country income equation |